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Correlated Equilibria and Communication in Games Françoise Forges Correlated equilibria and communication in games Françoise Forges To cite this version: Françoise Forges. Correlated equilibria and communication in games. Computational Complex- ity. Theory, Techniques, and Applications, pp.295-704, 2012, 10.1007/978-1-4614-1800-9_45. hal- 01519895 HAL Id: hal-01519895 https://hal.archives-ouvertes.fr/hal-01519895 Submitted on 9 May 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Correlated Equilibrium and Communication in Games Françoise Forges, CEREMADE, Université Paris-Dauphine Article Outline Glossary I. De…nition of the Subject and its Importance II. Introduction III. Correlated Equilibrium: De…nition and Basic Properties IV. Correlated Equilibrium and Communication V. Correlated Equilibrium in Bayesian Games VI. Related Topics and Future Directions VII. Bibliography Acknowledgements The author wishes to thank Elchanan Ben-Porath, Frédéric Koessler, R. Vijay Krishna, Ehud Lehrer, Bob Nau, Indra Ray, Jérôme Renault, Eilon Solan, Sylvain Sorin, Bernhard von Stengel, Tristan Tomala, Amparo Ur- bano, Yannick Viossat and, especially, Olivier Gossner and Péter Vida, for useful comments and suggestions. Glossary Bayesian game: an interactive decision problem consisting of a set of n players, a set of types for every player, a probability distribution which ac- counts for the players’ beliefs over each others’ types, a set of actions for every player and a von Neumann-Morgenstern utility function de…ned over n-tuples of types and actions for every player. Nash equilibrium: in an n-person strategic form game, a strategy n-tuple from which unilateral deviations are not pro…table. von Neumann-Morgenstern utility function: a utility function which re‡ects the individual’s preferences over lotteries. Such a utility function is de…ned over outcomes and can be extended to any lottery by taking expectation with respect to . 1 Pure strategy (or simply strategy): a mapping which, in an interac- tive decision problem, associates an action with the information of a player whenever this player can make a choice. Sequential equilibrium: a re…nement of the Nash equilibrium for n-person multistage interactive decision problems, which can be loosely de…ned as a strategy n-tuple together with beliefs over past information for every player, such that every player maximizes his expected utility given his beliefs and the others’strategies, with the additional condition that the beliefs satisfy (possibly sophisticated) Bayes updating given the strategies. Strategic (or normal) form game: an interactive decision problem con- sisting of a set of n players, a set of strategies for every player and a (typ- ically, von Neumann-Morgenstern) utility function de…ned over n-tuples of strategies for every player. Utility function: a real valued mapping over a set of outcomes which re- ‡ects the preferences of an individual by associating a utility level ( a “pay- o¤”) with every outcome. I. De…nition of the Subject and its Importance The correlated equilibrium is a game theoretic solution concept. It was proposed by Aumann (1974, 1987) in order to capture the strategic corre- lation opportunities that the players face when they take into account the extraneous environment in which they interact. The notion is illustrated in Section II. A formal de…nition is given in Section III. The correlated equilib- rium also appears as the appropriate solution concept if preplay communi- cation is allowed between the players. As shown in Section IV, this property can be given several precise statements according to the constraints imposed on the players’communication, which can go from plain conversation to ex- change of messages through noisy channels. Originally designed for static games with complete information, the correlated equilibrium applies to any strategic form game. It is geometrically and computationally more tractable than the better known Nash equilibrium. The solution concept has been extended to dynamic games, possibly with incomplete information. As an illustration, we de…ne in details the communication equilibrium for Bayesian games in Section V. 2 II. Introduction Example Consider the two-person game known as “chicken”,in which each player i can take a “paci…c”action (denoted as pi) or an “aggressive”action (denoted as ai): p2 a2 p1 (8; 8) (3; 10) a1 (10; 3) (0; 0) The interpretation is that player 1 and player 2 simultaneously choose an action and then get a payo¤, which is determined by the pair of chosen actions according to the previous matrix. If both players are paci…c, they both get 8. If both are aggressive, they both get 0. If one player is aggressive and the other is paci…c, the aggressive player gets 10 and the paci…c one gets 3. This game has two pure Nash equilibria (p1; a2), (a1; p2) and one mixed Nash equilibrium in which both players choose the paci…c action with 3 probability 5 , resulting in the expected payo¤ 6 for both players. A possible justi…cation for the latter solution is that the players make their choices as a function of independent extraneous random signals. The assumption of independence is strong. Indeed, there may be no way to prevent the players’ signals from being correlated. Consider a random signal which has no e¤ect on the players’ payo¤s and takes three possible values: low, medium or high, occurring each with 1 probability 3 . Assume that, before the beginning of the game, player 1 distinguishes whether the signal is high or not, while player 2 distinguishes whether the signal is low or not. The relevant interactive decision problem is then the extended game in which the players can base their action on the private information they get on the random signal, while the payo¤s only depend on the players’actions. In this game, suppose that player 1 chooses the aggressive action when the signal is high and the paci…c action otherwise. Similarly, suppose that player 2 chooses the aggressive action when the signal is low and the paci…c action otherwise. We show that these strategies form an equilibrium in the extended game. Given player 2’sstrategy, assume that player 1 observes a high signal. Player 1 deduces that the signal cannot be low so that player 2 chooses the paci…c action; hence player 1’sbest response is to play aggressively. Assume now that player 1 is informed that the signal 1 is not high; he deduces that with probability 2 , the signal is medium (i.e., 3 1 not low) so that player 2 plays paci…c and with probability 2 , the signal is low so that player 2 plays aggressive. The expected payo¤ of player 1 is 5:5 if he plays paci…c and 5 if he plays aggressive; hence, the paci…c action is a best response. The equilibrium conditions for player 2 are symmetric. To sum up, the strategies based on the players’private information form a Nash equilibrium in the extended game in which an extraneous signal is …rst selected. We shall say that these strategies form a “correlated equilibrium”. The corresponding probability distribution over the players’actions is p2 a2 1 1 1 p 3 3 (1) 1 1 a 3 0 and the expected payo¤ of every player is 7. This probability distribution can be used directly to make private recommendations to the players before the beginning of the game (see the canonical representation below). III. Correlated Equilibrium: De…nition and Basic Properties De…nition i i A game in strategic form G = (N; ( )i N ; (u )i N ) consists of a set of players N together with, for every player i 2 N, a set2 of strategies (for in- stance, a set of actions) i and a (von Neumann-Morgenstern)2 utility function i j u : R, where = j N is the set of all strategy pro…les. We assume ! 2 that the sets N and i, i N, are …nite. Q2 i A correlation device d = ( ; q; ( )i N ) is described by a …nite set of signals , a probability distributionPq over2 and a partition i of for every player i N. Since is …nite, the probability distributionPq is just a 2 real vector q = (q(!))! such that q(!) 0 and q(!) = 1. 2 ! 2 From G and d, we de…ne the extended game GXd as follows: ! is chosen in according to q every player i is informed of the element P i(!) of i which contains ! P G is played: every player i chooses a strategy i in i and gets the i j utility u (), = ( )j N . 2 4 i i A (pure) strategy for player i in Gd is a mapping : which is i i i i ! -measurable, namely, such that (!0) = (!) if !0 P (!). The inter- P 2 i pretation is that, in Gd, every player i chooses his strategy as a function of his private information on the random signal ! which is selected before the beginning of G. According to Aumann (1974), a correlated equilibrium of G is a pair i (d; ), which consists of a correlation device d = ( ; q; ( )i N ) and a Nash i P 2 equilibrium = ( )i N of Gd. The equilibrium conditions of every player i, conditionally on his private2 information, can be written as: i i i i i i q(!0 P (!))u ( (!0)) q(!0 P (!))u ( ; (!0)) (2) j j ! P i(!) ! P i(!) 02X 02X i N; i i; ! : q(!) > 0 8 2 8 2 8 2 i j where = ( )j=i.
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